Grade 10 Geometry Group

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Verify experimentally the properties of dilations given by a center and a scale factor:\n- A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.\n- The dilation of a line segment is longer or shorter in the ratio given by the scale factor. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Explain and use the relationship between the sine and cosine of complementary angles. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove the Laws of Sines and Cosines and use them to solve problems. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, Ö3) lies on the circle centered at the origin and containing the point (0, 2). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Derive the equation of a parabola given a focus and directrix. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Prove that all circles are similar. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Construct a tangent line from a point outside a given circle to the circle. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use permutations and combinations to compute probabilities of compound events and solve problems. In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). In Geometry we teach this with a clear model first, then enough practice that your child can explain the steps in their own words.